Given the equation p = 2 x y , find d y d t

Question

Given the equation   p  =            2      x        y  , find             d      y              d      t

Julius Haley · Accepted Answer

Using the properties of the logarithmic functions, it can be written that

\ln (a b) = \ln a - \ln b

.
It is known that

\frac{d}{d x} (\ln x) = \frac{1}{x} .

Use the properties of the logarithmic functions to simplify the function:

p = \frac{2 x}{y}

p = \frac{2 x}{y} y = \frac{2 x}{p} \ln y = \ln \frac{2 x}{p} = \ln 2 x - \ln p

Use the chain rule of derivative to differentiate both sides of the equation:

\ln y = \ln 2 x - \ln p

with respect to t and simplify to calculate

\frac{d y}{d t}

.

\ln y = \ln 2 x - \ln p \frac{d}{d t} (\ln y) = \frac{d}{d t} (\ln 2 x) - \ln p \frac{1}{y} \frac{d y}{d t} = \frac{1}{2 x} (2) \frac{d x}{d t} - \frac{1}{p} \frac{d p}{d t} = \frac{1}{x} \frac{d x}{d t} - \frac{1}{p} \frac{d p}{d t} \frac{d y}{d t} = y (\frac{1}{x} \frac{d x}{d t} - \frac{1}{p} \frac{d p}{d t}) = \frac{2 x}{p} (\frac{1}{x} \frac{d x}{d t} - \frac{1}{p} \frac{d p}{d t}) = \frac{2}{p} \frac{d x}{d t} - \frac{2 x}{p^{2}} \frac{d p}{d t}

Hence,

\frac{d y}{d t} = \frac{2}{p} \frac{d x}{d t} - \frac{2 x}{p^{2}} \frac{d p}{d t}

Given the equation p=(2x)/(y), find (dy)/(dt)